I have been asked to compute the seifert form of a knot, the twist knot.
I know how to compute the seifert surface, and then the seifert matrix seems to be defined accordingly (according to all the sources that I could find):
$V_{ij} = lk(x_i,x_j^+)$ (I believe this is called the linking number)
My problem is that I dont know what the $x_i$ and $x_j^+$ mean, Thank you very much in advance for answering those. May I add that I have been searching the internet for several hours with no clear explanation found, and that is why I'd be most grateful for your answer.
The Seifert surface $\Sigma\subset S^3$ is oriented, so the normal bundle $\nu$ is trivial; choose a nowhere-vanishing section $s\in \Gamma(\nu)$. For a curve $\gamma$, let $\gamma^+$ denote the curve obtained by pushing $\gamma$ off $\Sigma$ a small distance along $s$. Basically, we're taking a tubular neighborhood around $\Sigma$ and pushing $\gamma$ outward, but codimension $1$ and orientability mean that there's a well-defined direction in which to push at each point along $\gamma$. Descending to homology gives $\operatorname{lk}(x_i, x_j^+)$.